Taylor expansion displacement

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

The result in eq. (14) is not limited to the harmonic approximation because coefficients of like powers of the displacements on each side of eq. (10) are equal, irrespective of the order to which the Taylor expansions are made. 

Let us apply the Taylor expansion of the electronic matrix element to symmetry coordinates Qs (nuclear displacement coordinates) around a reference nuclear configuration Q0. Hence... 

It is necessary next to relate these small displacements to the collective coordinates g, from a Taylor expansion of the general potential energy U(Q). In the adiabatic approximation, the nuclear coordinates Q are free parameters and can be used as a basis for the Taylor expansion. Thus we write IJ(Q) in the general form (Ref. [2], Chapter 3) ... 

In this approach, the external potential displacements that are responsible for a transition from stage (i) to stage (ii) create conditions for the subsequent CT effects, in the spirit of the Born-Oppenheimer approximation. Clearly, the consistent second-order Taylor expansion at M°(co) does not involve the coupling hardness t A B and the off-diagonal response quantities of Eqs. (168) and (170), which vanish identically for infinitely separated reactants. However, since the interaction at Q modifies both the chemical potential difference and the... 

According to (42), the T2 mode is not JT active (in first order) in E states. However, the matrix elements of the SO operator with non-relativistic electronic wave functions vanish for a state in symmetry. It is then essential to take account of the leading nonvanishing terms in the Taylor expansion of the matrix elements of the SO operator. As shown below, these are of first order of vibrational displacements of T2 symmetry, which implies the existence of a purely relativistic E xT JT effect [34]. Linear E xT vibronic coupling is not accounted for by the JT selection rules [2] it is, therefore, a novel type of JT effect. 

The relevant second-order Taylor expansion of the molecular electronic energy in powers of displacements of the canonical state parameters, [d/V, dl/(r)], is determined by the relevant principal derivatives of the energy representation ... 

If the off-diagonal matrix element is non-zero, the two roots are different they correspond to a lower (upper) sheet of the adiabatic potential surface for which a crossover is excluded. In order to show that Hx2 i1 0 let us apply the Taylor expansion of that matrix element with respect to symmetry coordinates Qs (nuclear displacement coordinates) around a reference nuclear configuration Q0, hence... 

Natural collision coordinates are clearly hopeless. They are based firmly on one path (which one to choose ) and will probably be undefined, due to path curvature, at large distances from the defining path, where other important paths may reside. Moreover, a Taylor expansion in Cartesian (or MWC) displacements from one path will not give us a PES that is properly symmetric with respect to the CNPI group or even symmetric under rotation and inversion of the molecule. 

In order to relate CS of reactants A and B to the CT reaction rate an intersecting-state model (ISM) has recently been proposed [36], with the relevant potential energy curves defined in the electron population space. We again consider the reactive system = A—B and the associated potential energy surface Ejj(Na, Nb), given by the second-order Taylor expansion in the reactant population displacements from the initial configuration, = (A (B ), before CT (see Figs. 6 and 7) ... 

From this time on, all other basins of the function V (/ ) have disappearedfrom the theory -only motion in the neighborhood of Rq is to be considered. If someone is aiming to apply harmonic approximation and to consider small displacements from Rq (as we do), then it is a good idea to write down the Taylor expansion of V about Rq [hereafter, instead of the symbols X, Yi, Z, X2,Y2, Z2. for the atomic Cartesian coordinates, we will use a slightly more uniform notation R = (Zi, X2, Z3, Z4, Z5, Xe,. .Zsjv) ] ... 

H,y = 3 V /dxjdxj (i and j run from 1 to 3A/). The Hjt is the curvature of the potential energy surface at the point in configuration space, which with the slope vector allows the energy V at a slightly displaced point to be expressed as a Taylor expansion... 

There remain the small, homogeneous reactors. For such reactors the Taylor expansion which breaks off at the third term is invalid— partly because the kernels are no longer displacement kernels, partly because the Taylor series does not converge well. The second consideration suggests that the higher moments of the kernel G x — A ) are important—i.e., that for small reactors, the full spatial dependence of the kernels must be considered. 

The basis for the harmonic model is the assumption that the molecules in the solid have well-defined equilibrium positions and orientations, so that one can make a Taylor expansion of the potential about these positions and orientations and truncate after the quadratic terms. It is not always realized that the rotational part of the kinetic energy, Eq. (13), must be approximated too, see Ref. [27]. The dynamical coordinates for the molecular rotations in the harmonic model are linearized angular displacements AQ = (AQ, AQy, AQJ. Together with the center of mass displacements ii = (m, m )... 

The dependence of the intermolecular potential Vab on the intramolecular vibrational coordinates can be made explicit by writing it as a Taylor expansion in the atomic displacement coordinates... 

The above expressions are the same as eqn (9.3), but reflect the system in two states. In eqn (9.9), is the gradient of the energy with respect to the peripheral coordinates of the reactant state, while gi is the gradient of the energy with respect to the interior coordinates of the reactant state. Similar definitions hold for the transition state gradients. Within the Taylor expansion for eqns (9.9) and (9.10), the reactant minimum and the corresponding saddle point for any peripheral coordinate displacement, can be re-optimized by using the interior coordinate steps ... 

The equilibrium density is a functional of the field n(i) that is, if we specify vfunctional Taylor expansion gives... 

Assuming that we can do a Taylor expansion of the dependence of a molecular property P on the nuclear displacements along the normal coordinates Q, we may... 

In the following text we will denote an equilibrium value by using an arc over the relevant species. Please note the different meaning of compared with the previous paragraphs. There the same symbol denoted the energy of the bonding level at a given nuclear distance, while here it indicates the special value at the equilibrium distance. The Taylor expansion of the expression of Eq. (2.20a) yields a harmonic behaviour (Hooke s law) for small displacements (r F) (see Fig. 2.3) ... 

A displacement of s(A) by an amount along the z axis may be written as a Taylor expansion in the displacement ... 

Taylor s series in terms of the normal coordinate Qi (Appendix). If the displacement is small we may ignore higher order terms in the expansion. 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

The electronic energy of a molecule, ion, or radical at geometries near a stable structure can be expanded in a Taylor series in powers of displacement coordinates as was done in the preceding section of this Chapter. This expansion leads to a picture of uncoupled harmonic vibrational energy levels... 

The small parameter of the expansion is the mode displacement SQi, because it is always evaluated for the mode ground state or a low-energy state. Both are very localized. Let us consider the eigenvector Qi of the mode i. Then the Hamiltonian can be expanded in a Taylor series on the displacements 5Qi. 

---